Yukio is very mentally strong. She doesn't take Kuros bullshit and is able to handle her child well. Just because she can't defeat literal demons and can do chores that she had to learn growing up in that culture doesn't mean she isn't strong. Also what is wrong with not working in the workplace? Women are now taking careers outside workplaces so I don't know why your putting that so high up. Can you not be hardworking if it's not in a workplace?
In conclusion: if you don't think a single mother struggling to make ends meet that bounces back whenever someone awful happens to her isn't strong you do not know what strong is. You have to be fucking strong to take care of a child alone and you have to be fucking strong to do all the cleaning, cooking, and working in a house for 3 years having a low paying job. She asked Kuro to be the babysitter because of how fucking hard it is to be a single mother.
To be honest, with the strong female lead tag, I expected Yuki to be much stronger than she is in this manga. Upon reading the description, I thought Yuki made Kuros her kid's babysitter because she wanted to focus on her career. But her career isn't even at a workplace... -_-. Kuros can't even cook or babysit, and in later chapters, Yuki is the one cooking for her kid and Kuros. Yuki can't even protect herself.